Some estimates of Schrödinger type operators on variable Lebesgue and Hardy spaces

Abstract

In this article, the authors consider the Schrödinger type operator $L:=-\mathrm{div}(A\nabla )+V$ on ${\mathbb {R}}^n$ with $n\ge 3$, where the matrix A satisfies uniformly elliptic condition and the nonnegative potential V belongs to the reverse Hölder class $RH_q({\mathbb {R}}^n)$ with $q\in (n/2,\,\infty )$. Let $p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\,\infty )$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition. When $p(\cdot ):\ {\mathbb {R}}^n\rightarrow (1,\,\infty )$, the authors prove that the operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla ^2L^{-1}$ are bounded on variable Lebesgue space $L^{p(\cdot )}({\mathbb {R}}^n)$. When $p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\,1]$, the authors introduce the variable Hardy space $H_L^{p(\cdot )}({\mathbb {R}}^n)$, associated to L, and show that $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla ^2L^{-1}$ are bounded from $H_L^{p(\cdot )}({\mathbb {R}}^n)$ to $L^{p(\cdot )}({\mathbb {R}}^n)$.